Another good virus would be a Bacteriophage, a virus that infects and kills illness-causing bacteria. From Wiki:

A bacteriophage also known informally as a phage, is a virus that infects and replicates within Bacteria and Archaea. The term was derived from "bacteria" and the Greek φαγεῖν (phagein), "to devour". Bacteriophages are composed of proteins that encapsulate a DNA or RNA genome, and may have relatively simple or elaborate structures.

They have been used for over 90 years as an alternative to antibiotics in the former Soviet Union and Central Europe as well as in France. They are seen as a possible therapy against multi-drug-resistant strains of many bacteria.

Intentionally using Bacteriophages medicinally is called phage therapy.

I would say that if any "good" viruses exist, they are already within us. Retrotransposons are genetic elements in our DNA that were likely ancient viruses and they move around from time to time either by excising themselves and moving somewhere else or by making a copy and inserting it somewhere else in the genome. Even though we are born with them, their activity is similar to modern viruses. When retrotransposons insert themselves in a new place they can cause disease, but the variation they cause likely brought us some beneficial advantage at some point because we have so many of them and they stuck around this long.

We have engineered a few good viruses to treat certain diseases

Per my comment and response:

The most current example (at this time and based on my recollection) is the virus we have engineered to treat a certain type of macro degenerative eye condition:

Scientists Have Reversed Age-Related Blindness by Deliberately Infecting Eyes With a Virus

There are a few other cases out there, and, as always, xkcd is there to help:

Cowpox and smallpox viruses structurally similar, and catching one confers immunity to both by immune system response, but one was a deadly disease and the other almost harmless. Once this was discovered, the days of smallpox were numbered. We had the means and the motivation to stamp it out.

On my last check a few years ago, we are deliberately keeping cowpox alive to ensure that we can kill smallpox should it ever come back.

The virus that ended a plague is a good virus.

Biologically, all viruses by themselves are bad. More accurately, they have a negative effect. They work by injecting a strand of DNA into one of the host's cells, which makes the cell produce viruses instead of more cells. However, some viruses negative effect when paired with another virus can produce a double negative. Unfortunately, it never quite cancels out.

To sum it up, if you have a virus, there is definitely something bad happening in your system.

So, do such beneficial for their direct hosts viruses exist? If so, what are they called? What are the examples?

Bacterial viruses, termed bacteriophages or simply phages are normally killers of bacteria and thus occasionally useful for humans (e.g. phage therapy) but not beneficial for their hosts. However, some phages, called temperate phages, have a lysogenic life cycle, where the viral genome can become integrated into the bacterial chromosome and they replicate with the bacteria (at least temporarily) as a prophage. These prophages have a shared evolutionary fate with the bacteria and can be directly beneficial by carrying bacterial fitness factors such as virulence factors or antibiotic resistance genes. You might be interested to read more in our short review.